Constructing Majorana Representations

TL;DR

This paper introduces an algorithm for constructing Majorana representations of finite groups, enabling the discovery of new Majorana algebras and advancing the study of structures related to the Monster group.

Contribution

It presents a novel algorithm for constructing Majorana representations, expanding the catalog of known Majorana algebras and providing new tools for algebraic research.

Findings

Constructed Majorana representations for several groups including new examples.

Identified new Majorana algebras through the developed algorithm.

Enhanced understanding of the relationship between groups and Majorana structures.

Abstract

Majorana theory was introduced by A. A. Ivanov as an axiomatic framework in which to study objects related to the Monster simple group and the Griess algebra. Since its inception, it has been used to construct a number of new and important subalgebras of the Griess algebra. The objects at the centre of the theory are known as Majorana algebras and can be studied either in their own right, or as Majorana representations of finite groups. In this paper, we present an algorithm to construct the Majorana representations of a given group. We also list a number of groups for which we have constructed Majorana representations, including some which give new examples of Majorana algebras. This work is inspired by that of A. Seress.